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Sound transmission in two-dimensional periodic poroelastic structures

Qiao Hou He Zeng Zhang Heng-Kun Peng Wei-Cai Jiang Wen

Sound transmission in two-dimensional periodic poroelastic structures

Qiao Hou, He Zeng, Zhang Heng-Kun, Peng Wei-Cai, Jiang Wen
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  • To obtain excellent sound reduction performance, in this paper we introduce a novel periodic poroelastic composite structure, which combines poroelastic material and periodic structure and aims at using the remarkable acoustic performance of these two. This periodic composite structure comprises three parts, i.e. the poroelastic domain, the elastic domain (thin plate), and the periodic resonators, which can be simple single-degree-of-freedom resonators (SRs) or composite two-degree-of-freedom resonators (CRs). A theoretical model is established by using Biot theory for the poroelastic domain, and by using the effective medium method for the resonator-plate coupling system, which is considered as an isotropic plate with an equivalent dynamic density. This method is validated with degenerated model in the literature; the results obtained by this method are in excellent consistence with the results in the literature. Parameter analyses are performed to test the influences of poroelastic addition and periodic resonator on the sound transmission loss (STL) of this periodic composite structure under two kinds of boundary conditions. The poroelastic addition is found to increase the STL while the influences of resonators are complicated. The STL increases notably in the frequency range bounded by the characteristic frequencies of these resonators, however, a decrease just follows when it exceeds these frequencies, which can be observed in both SR case and CR case under the two boundary conditions. In the meantime, when multiple SR is placed in a periodic lattice, it is found that different resonators with ascending mass and characteristic frequencies have superior STL to those with ascending characteristic frequencies but have equal mass. The case with CR, which is more complicated as expected, shows less STL decrease than the case with SR, but wider frequency range where the STL increases than a poroelastic composite structure without resonators. This results from the fact that the frequency band of vibration suppression in the CR case is wider than in the SR case. As a result, to achieve the desired STL performance in a frequency range, the proposed composite structure using SR with tuned characteristic frequencies is enough; however, if a wider frequency band is expected even if there is a slight STL tradeoff, the CR case is a better option. Though the method proposed is only valid in the low-to-medium frequency range, the results obtained can benefit theoretical development of low-to-medium sound modulation applications, they are also valuable and illuminating for investigating the broadband sound modulation.
      Corresponding author: Jiang Wen, wjiang@hust.edu.cn
    [1]

    Allard J F, Atalla N 2009 Propagation of Sound in Porous Media (Chichester: John Wiley & Sons) pp111−134

    [2]

    Deckers E, Jonckheere S, Vandepitte D, Desmet W 2015 Arch. Comput. Methods Eng. 22 183

    [3]

    Bolton J S, Shiau N M, Kang Y J 1996 J. Sound Vib. 191 317

    [4]

    Zhou J, Bhaskar A, Zhang X 2013 J. Sound Vib. 332 3724

    [5]

    Liu Y 2015 J. Sound Vib. 339 376

    [6]

    Qiao H, He Z, Jiang W, Peng W 2019 J. Sound Vib. 440 256

    [7]

    Allard J F, Depollier C, Rebillard P, Lauriks W, Cops A 1989 J. Appl. Phys. 66 2278

    [8]

    Deresiewicz H 1961 Bull. Seismol. Soc. Am. 51 51

    [9]

    Zhou J, Bhaskar A, Zhang X 2013 Appl. Acoust. 74 1422

    [10]

    Liu Y, Sebastian A 2015 J. Sound Vib. 344 399

    [11]

    Talebitooti R, Daneshjou K, Kornokar M 2016 J. Sound Vib. 363 380

    [12]

    Shojaeifard M H, Talebitooti R, Ranjbar B, Ahmadi R 2014 Appl. Math. Mech. 35 1447

    [13]

    Schanz M 2009 Appl. Mech. Rev. 62 030803

    [14]

    Panneton R, Atalla N 1997 J. Acoust. Soc. Am. 101 3287

    [15]

    Verdière K, Panneton R, Elkoun S, Dupont T, Leclaire P 2013 J. Acoust. Soc. Am. 134 4648

    [16]

    Brillouin L 2003 Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices (New York: Dover Publications) pp1−16

    [17]

    Mead D M 1996 J. Sound Vib. 190 495

    [18]

    曹永军, 周培勤, 董纯红 2006 物理学报 55 6470

    Cao Y J, Zhou P Q, Dong C H 2006 Acta Phys. Sin. 55 6470

    [19]

    丁昌林, 董仪宝, 赵晓鹏 2018 物理学报 67 194301

    Ding C L, Dong Y B, Zhao X P 2018 Acta Phys. Sin. 67 194301

    [20]

    温熙森 2006 光子/声子晶体理论与技术 (北京: 科学出版社) 第38−341页

    Wen X S 2006 Photonic/Phononic Theory and Technology (Beijing: Science Press) pp38−104 (in Chinese)

    [21]

    姜久龙, 姚宏, 杜军, 赵静波, 邓涛 2017 物理学报 66 064301

    Jiang J L, Yao H, Du J, Zhao J B, Deng T 2017 Acta Phys. Sin. 66 064301

    [22]

    林国昌, 孙宏伟, 谭惠丰, 杜星文 2011 物理学报 60 034302

    Lin G C, Sun H W, Tan H F, Du X W 2011 Acta Phys. Sin. 60 034302

    [23]

    丁昌林, 赵晓鹏 2009 物理学报 58 6351

    Ding C L, Zhao X P 2009 Acta Phys. Sin. 58 6351

    [24]

    Yu D L, Shen H J, Liu J W, Yin J F, Zhang Z F, Wen J H 2018 Chin. Phys. B 27 064301

    [25]

    张丰辉, 唐宇帆, 辛锋先, 卢天健 2018 物理学报 67 234302

    Zhang F H, Tang Y F, Xin F X, Lu T J 2018 Acta Phys. Sin. 67 234302

    [26]

    朱席席, 肖勇, 温激鸿, 郁殿龙 2016 物理学报 65 176202

    Zhu X X, Xiao Y, Wen J H, Yu D L 2016 Acta Phys. Sin. 65 176202

    [27]

    Chen X, Cai L, Wen J H 2018 Chin. Phys. B 27 057803

    [28]

    宋玉宝, 温激鸿, 郁殿龙, 沈惠杰 2018 机械工程学报 54 60

    Song Y B, Wen J H, Yu D L, Shen H J 2018 Chin. J. Mech. Eng. 54 60

    [29]

    Kidner M R F, Fuller C R, Gardner B 2006 J. Sound Vib. 294 466

    [30]

    Idrisi K, Johnson M E, Toso A, Carneal J P 2009 J. Sound Vib. 323 51

    [31]

    Cui S, Harne R L 2017 J. Acoust. Soc. Am. 141 4715

    [32]

    Harne R L, Song Y, Dai Q 2017 Extrem. Mech. Lett. 12 41

    [33]

    Bishop J, Harne R L 2018 Appl. Acoust. 130 222

    [34]

    温激鸿, 郁殿龙, 赵宏刚, 蔡力, 肖勇, 王刚, 尹剑飞 2015 人工周期结构中弹性波的传播: 振动与声学特性 (北京: 科学出版社) 第272−290页

    Wen J H, Yu D L, Zhao H G, Cai L, Xiao Y, Wang G, Yin J F 2016 Propagation of Elastic Waves in Artificial Periodic Structures: Vibrational and Acoustical Properties (Beijing: Science Press) pp272−290 (in Chinese)

    [35]

    Xiao Y, Wen J, Wen X 2012 J. Sound Vib. 331 5408

    [36]

    Li P, Yao S, Zhou X, Huang G, Hu G 2014 J. Acoust. Soc. Am. 135 1844

    [37]

    Xiao Y, Wen J, Wen X 2012 J. Phys. D: Appl. Phys. 45 195401

    [38]

    Biot M A 1956 J. Acoust. Soc. Am. 28 179

    [39]

    Junger M C, Feit D 1986 Sound, Structures, and Their Interaction (Massachusetts: MIT Press) pp235−277

    [40]

    程建春 2012 声学原理 (北京: 科学出版社) 第51页

    Cheng J C 2012 Theory of Sound (Beijing: Science Press) p51 (in Chinese)

    [41]

    Den Hartog J P 1985 Mechanical Vibrations (New York: Dover Publications) pp79−121

    [42]

    何琳, 朱海潮, 邱小军, 杜功焕 2006 声学理论与工程应用 (北京: 科学出版社) 第173−184页

    He L, Zhu H C, Qiu X J, Du G H 2006 Theory and Engineering Applications of Acoustics (Beijing: Science Press) pp173−184 (in Chinese)

    [43]

    Peng H, Frank Pai P, Deng H 2015 Int. J. Mech. Sci. 103 104

  • 图 1  含多孔介质复合结构及其子结构示意图 (a) 含多孔介质复合结构; (b) 等效模型; (c) OU边界; (d) OB边界; (e) 板受力情况(OU边界); (f) 板受力情况(OB边界); (g) 弹簧振子受力

    Figure 1.  Schematic of the poroelastic composite structure and its substructures: (a) The poroelastic composite structure; (b) the equivalent model; (c) the OU boundary connection; (d) the OB boundary connection; (e) the forces in OU boundary case; (f) the forces in OB boundary case; (g) the forces in a simple spring-mass resonator.

    图 2  组合振子系统示意图 (a) 串联弹簧振子系统A; (b) 复合弹簧振子系统B

    Figure 2.  Schematic of the composite-resonator-structure: (a) Composite resonator type A, two resonators placed in serial connection; (b) composite resonator type B, two resonators placed in composite connection.

    图 3  周期振子排布方式示意图 (a) 简单振子周期分布, 按各个振子质量${m_i}$和特征频率${f_i}$分为多个振子情况 (N1SR, ${m_i}$${f_i}$均保持恒定) 和多种振子情况 (NNSR, ${m_i}$${f_i}$不相同); (b) 组合振子周期分布, 按振子部件质量$m_n^i$和特征频率$f_n^i$分为多个振子情况 (N1CR, $m_n^i$$f_n^i$均保持恒定) 和多种振子情况 (NNCR, $m_n^i$$f_n^i$不相同); 图中虚线框内部分为单个振子单元, (b)中虚线框部分可替换为 图2中B类组合振子

    Figure 3.  Schematic of the arrangement of periodic resonators: (a) An array of simple resonators, denoted as multiple resonators (N1SR, with constant ${m_i}$ and ${f_i}$) or multiple kinds of resonators (NNSR, with different ${m_i}$ and ${f_i}$); (b) an array of composite resonators, denoted as multiple resonators (N1CR, with constant $m_n^i$ and $f_n^i$) or multiple kinds of resonators (NNCR, with different $m_n^i$ or $f_n^i$). The area in the dash-line denotes the periodic lattice, in panel (b), the composite resonator can be type B in Fig. 2

    图 4  不同类型隔声结构验证算例 (a) 文献[35]随机入射情况; (b) 文献[35]斜入射情况; (c) 文献[3]含多孔介质复合结构; 其中, 各曲线为本文结果, 各标记为文献中结果

    Figure 4.  Validation of the results here with previous results: (a) The diffuse case in Ref. [35]; (b) the oblique incident cases in Ref. [35]; (c) the composite poroelastic structure without resonator in Ref. [3]. The lines are results obtained here, while the marks are the results in the references.

    图 5  有无多孔材料对含不同特征频率振子系统复合结构STL的影响 (a) OU边界情况; (b) OB边界情况; 有无多孔介质分别与相应实线和虚线对应

    Figure 5.  Influence of porous material on the STL of the multiple-single-type-resonator composite structure with different characteristic frequencies: (a) OU case; (b) OB case. The solid lines correspond to cases with porous materials.

    图 6  含相同简单振子系统复合结构(fr = 300 Hz)有无多孔介质及相应不含振子复合结构的STL (有多孔介质, Porous + Resonator; 无多孔介质, Resonator; 相应不含振子复合结构, Porous) (a) OU边界情况; (b) OB边界情况.

    Figure 6.  The STL of multiple-single-type-resonator composite structure (fr = 300 Hz) with/without porous, and composite structure without resonators: (a) OU case; (b) OB case. Composite structure here with porous material: Porous + Resonator. Without porous material: Resonator. Composite structure without resonators: Porous.

    图 7  采用不同特征频率简单振子系统对复合结构STL的影响 (a) OU边界; (b) OB边界

    Figure 7.  Influences of resonators with different characteristic frequencies on the STL: (a) OU case; (b) OB case.

    图 8  单一类型简单振子周期排布时 (a) OU, OB情况下STL及其位移传递率Ti; (b) 振子的等效质量meq和板等效动态密度${\rho _{{\rm{eq}}}}$

    Figure 8.  (a) STL of OU and OB case in periodically-arranged single simple resonator case, and its displacement transmissibility Ti; (b) equivalent mass meq of a single resonator and the dynamic density ${\rho _{{\rm{eq}}}}$ of the equivalent plate.

    图 9  两类组合振子系统中质量块的位移传递率${T_1}$, ${T_2}$和动态质量${m_{{\rm{eq}}}}$ (a1) 组合振子系统A中各质量块的位移传递率${T_1}$, ${T_2}$; (a2) 组合振子系统A的动态质量${m_{{\rm{eq}}}}$; (b1) 组合振子系统B中各质量块的位移传递率${T_1}$, ${T_2}$; (b2) 组合振子系统B的动态质量${m_{{\rm{eq}}}}$

    Figure 9.  Displacement transmissibility and dynamic mass of the mass components in the two composite resonators: (a1) Displacement transmissibility ${T_1}$, ${T_2}$ of composite resonator type A; (a2) dynamic mass ${m_{{\rm{eq}}}}$ of composite resonator type A; (b1) displacement transmissibility ${T_1}$, ${T_2}$ of composite resonator type B; (b2) dynamic mass ${m_{{\rm{eq}}}}$ of composite resonator type B.

    图 10  复合结构周期间隔内分布4个相同简单振子(Single resonator), 组合振子A或组合振子B时的STL和不含振子复合结构(Without resonator)的STL (a) OU边界情况; (b) OB边界情况

    Figure 10.  STL of the proposed composite structure with 4 identical simple resonators (Single resonator), composite resonators of type A or B versus its STL without any resonators (Without resonator) in a periodic lattice: (a) OU boundary case; (b) OB boundary case.

    图 11  NNSR分布时OU, OB边界情况下的STL (a1), (a2) 情况A; (b1), (b2) 情况B, ${{\Delta m} / {{m_{{\rm{sum}}}}}} = 0.04$; 其中, (a1)和(b1)为OU边界情况, (a2)和(b2)为OB边界情况

    Figure 11.  STL of the composite structure with NNSR configuration under two boundary cases: (a1), (a2) Case A; (b1), (b2) case B, ${{\Delta m} / {{m_{{\rm{sum}}}}}} = 0.04$. Here (a1) and (b1) correspond to OU case, (a2) and (b2) correspond to OB case.

    图 12  采用组合振子复合结构的STL (a1), (a2) 采用组合振子A; (b1), (b2) 采用组合振子B; 其中, (a1)和(b1)对应于OU边界情况, (a2)和(b2)对应于OB边界情况

    Figure 12.  STL of the proposed composite structure under NNCR configuration: (a1), (a2) Composite resonator type A; (b1), (b2) composite resonator type B. Here (a1) and (b1) correspond to OU case, (a2) and (b2) correspond to OB case.

    图 13  不同振子系统分布时STL对比 (a), (a1) OU边界情况; (b), (b1) OB边界情况; $\Delta m = 0$$\Delta m > 0$对应简单振子情况NNSR; Type A和Type B对应组合振子情况NNCR

    Figure 13.  STL of different resonator system configuration: (a), (a1) OU case; (b), (b1) OB case. $\Delta m = 0$ and $\Delta m > 0$ correspond to simple resonator case NNSR. Type A and Type B correspond to composite resonator case NNCR.

    表 1  振子系统分布情况简称及其对应含义

    Table 1.  Abbreviations of the distribution of resonator systems and their meanings.

    振子系统分布含义
    N1SR周期间隔内分布多个简单振子系统, 各振子系统${m_i}$和${f_i}$均相等
    NNSR周期间隔内分布多种简单振子系统, 各振子系统${m_i}$或${f_i}$不同
    N1CR周期间隔内分布多个组合振子系统, 各振子系统$m_n^i$和$f_n^i$均相等
    NNCR周期间隔内分布多种组合振子系统, 各振子系统$m_n^i$或$f_n^i$不同
    DownLoad: CSV

    表 2  薄板、空气域及振子参数

    Table 2.  Parameters of plate and resonators.

    薄板 空气域(20 ℃, 1 atm) 简单振子
    h/mmE/GPaν${\rho _{\rm{p}}}$/kg·m–3 ρ/kg·m–3c0/m·s–1ha/mm fr/Hzγ
    1700.332700 1.204343.212 3000.2
    DownLoad: CSV

    表 3  多孔材料参数

    Table 3.  Parameters of the porous media used here

    参数参数
    ${\rho _{\rm{s}}}$/kg·m–330 $\epsilon$0.9
    ${\rho _{\rm{f}}}$/kg·m–31.204 ${\tau _\infty }$7.8
    ${E_{\rm{s}}}$/MPa0.8(1+0.265${\rm{j}}$) ${\sigma _0}$/MKS rayls25000
    ${\nu _s}$0.4 hp/mm50
    DownLoad: CSV

    表 4  振子系统参数

    Table 4.  Parameters of the composite resonators.

    参数$\omega _1^i$$\omega _2^i$rs$\eta _1^i$$\eta _2^i$
    600${\text{π}}$10880.0750.06250.010.05
    DownLoad: CSV

    表 5  组合振子系统参数

    Table 5.  Parameters of the composite resonators.

    参数rs$\eta _1^i$$\eta _2^i$
    0.0450.0400.010.05
    DownLoad: CSV
  • [1]

    Allard J F, Atalla N 2009 Propagation of Sound in Porous Media (Chichester: John Wiley & Sons) pp111−134

    [2]

    Deckers E, Jonckheere S, Vandepitte D, Desmet W 2015 Arch. Comput. Methods Eng. 22 183

    [3]

    Bolton J S, Shiau N M, Kang Y J 1996 J. Sound Vib. 191 317

    [4]

    Zhou J, Bhaskar A, Zhang X 2013 J. Sound Vib. 332 3724

    [5]

    Liu Y 2015 J. Sound Vib. 339 376

    [6]

    Qiao H, He Z, Jiang W, Peng W 2019 J. Sound Vib. 440 256

    [7]

    Allard J F, Depollier C, Rebillard P, Lauriks W, Cops A 1989 J. Appl. Phys. 66 2278

    [8]

    Deresiewicz H 1961 Bull. Seismol. Soc. Am. 51 51

    [9]

    Zhou J, Bhaskar A, Zhang X 2013 Appl. Acoust. 74 1422

    [10]

    Liu Y, Sebastian A 2015 J. Sound Vib. 344 399

    [11]

    Talebitooti R, Daneshjou K, Kornokar M 2016 J. Sound Vib. 363 380

    [12]

    Shojaeifard M H, Talebitooti R, Ranjbar B, Ahmadi R 2014 Appl. Math. Mech. 35 1447

    [13]

    Schanz M 2009 Appl. Mech. Rev. 62 030803

    [14]

    Panneton R, Atalla N 1997 J. Acoust. Soc. Am. 101 3287

    [15]

    Verdière K, Panneton R, Elkoun S, Dupont T, Leclaire P 2013 J. Acoust. Soc. Am. 134 4648

    [16]

    Brillouin L 2003 Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices (New York: Dover Publications) pp1−16

    [17]

    Mead D M 1996 J. Sound Vib. 190 495

    [18]

    曹永军, 周培勤, 董纯红 2006 物理学报 55 6470

    Cao Y J, Zhou P Q, Dong C H 2006 Acta Phys. Sin. 55 6470

    [19]

    丁昌林, 董仪宝, 赵晓鹏 2018 物理学报 67 194301

    Ding C L, Dong Y B, Zhao X P 2018 Acta Phys. Sin. 67 194301

    [20]

    温熙森 2006 光子/声子晶体理论与技术 (北京: 科学出版社) 第38−341页

    Wen X S 2006 Photonic/Phononic Theory and Technology (Beijing: Science Press) pp38−104 (in Chinese)

    [21]

    姜久龙, 姚宏, 杜军, 赵静波, 邓涛 2017 物理学报 66 064301

    Jiang J L, Yao H, Du J, Zhao J B, Deng T 2017 Acta Phys. Sin. 66 064301

    [22]

    林国昌, 孙宏伟, 谭惠丰, 杜星文 2011 物理学报 60 034302

    Lin G C, Sun H W, Tan H F, Du X W 2011 Acta Phys. Sin. 60 034302

    [23]

    丁昌林, 赵晓鹏 2009 物理学报 58 6351

    Ding C L, Zhao X P 2009 Acta Phys. Sin. 58 6351

    [24]

    Yu D L, Shen H J, Liu J W, Yin J F, Zhang Z F, Wen J H 2018 Chin. Phys. B 27 064301

    [25]

    张丰辉, 唐宇帆, 辛锋先, 卢天健 2018 物理学报 67 234302

    Zhang F H, Tang Y F, Xin F X, Lu T J 2018 Acta Phys. Sin. 67 234302

    [26]

    朱席席, 肖勇, 温激鸿, 郁殿龙 2016 物理学报 65 176202

    Zhu X X, Xiao Y, Wen J H, Yu D L 2016 Acta Phys. Sin. 65 176202

    [27]

    Chen X, Cai L, Wen J H 2018 Chin. Phys. B 27 057803

    [28]

    宋玉宝, 温激鸿, 郁殿龙, 沈惠杰 2018 机械工程学报 54 60

    Song Y B, Wen J H, Yu D L, Shen H J 2018 Chin. J. Mech. Eng. 54 60

    [29]

    Kidner M R F, Fuller C R, Gardner B 2006 J. Sound Vib. 294 466

    [30]

    Idrisi K, Johnson M E, Toso A, Carneal J P 2009 J. Sound Vib. 323 51

    [31]

    Cui S, Harne R L 2017 J. Acoust. Soc. Am. 141 4715

    [32]

    Harne R L, Song Y, Dai Q 2017 Extrem. Mech. Lett. 12 41

    [33]

    Bishop J, Harne R L 2018 Appl. Acoust. 130 222

    [34]

    温激鸿, 郁殿龙, 赵宏刚, 蔡力, 肖勇, 王刚, 尹剑飞 2015 人工周期结构中弹性波的传播: 振动与声学特性 (北京: 科学出版社) 第272−290页

    Wen J H, Yu D L, Zhao H G, Cai L, Xiao Y, Wang G, Yin J F 2016 Propagation of Elastic Waves in Artificial Periodic Structures: Vibrational and Acoustical Properties (Beijing: Science Press) pp272−290 (in Chinese)

    [35]

    Xiao Y, Wen J, Wen X 2012 J. Sound Vib. 331 5408

    [36]

    Li P, Yao S, Zhou X, Huang G, Hu G 2014 J. Acoust. Soc. Am. 135 1844

    [37]

    Xiao Y, Wen J, Wen X 2012 J. Phys. D: Appl. Phys. 45 195401

    [38]

    Biot M A 1956 J. Acoust. Soc. Am. 28 179

    [39]

    Junger M C, Feit D 1986 Sound, Structures, and Their Interaction (Massachusetts: MIT Press) pp235−277

    [40]

    程建春 2012 声学原理 (北京: 科学出版社) 第51页

    Cheng J C 2012 Theory of Sound (Beijing: Science Press) p51 (in Chinese)

    [41]

    Den Hartog J P 1985 Mechanical Vibrations (New York: Dover Publications) pp79−121

    [42]

    何琳, 朱海潮, 邱小军, 杜功焕 2006 声学理论与工程应用 (北京: 科学出版社) 第173−184页

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  • Received Date:  28 January 2019
  • Accepted Date:  03 April 2019
  • Available Online:  16 August 2019
  • Published Online:  01 June 2019

Sound transmission in two-dimensional periodic poroelastic structures

    Corresponding author: Jiang Wen, wjiang@hust.edu.cn
  • 1. Department of Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China
  • 2. National Key Laboratory on Ship Vibration and Noise, China Ship Development and Design Center, Wuhan 430064, China

Abstract: To obtain excellent sound reduction performance, in this paper we introduce a novel periodic poroelastic composite structure, which combines poroelastic material and periodic structure and aims at using the remarkable acoustic performance of these two. This periodic composite structure comprises three parts, i.e. the poroelastic domain, the elastic domain (thin plate), and the periodic resonators, which can be simple single-degree-of-freedom resonators (SRs) or composite two-degree-of-freedom resonators (CRs). A theoretical model is established by using Biot theory for the poroelastic domain, and by using the effective medium method for the resonator-plate coupling system, which is considered as an isotropic plate with an equivalent dynamic density. This method is validated with degenerated model in the literature; the results obtained by this method are in excellent consistence with the results in the literature. Parameter analyses are performed to test the influences of poroelastic addition and periodic resonator on the sound transmission loss (STL) of this periodic composite structure under two kinds of boundary conditions. The poroelastic addition is found to increase the STL while the influences of resonators are complicated. The STL increases notably in the frequency range bounded by the characteristic frequencies of these resonators, however, a decrease just follows when it exceeds these frequencies, which can be observed in both SR case and CR case under the two boundary conditions. In the meantime, when multiple SR is placed in a periodic lattice, it is found that different resonators with ascending mass and characteristic frequencies have superior STL to those with ascending characteristic frequencies but have equal mass. The case with CR, which is more complicated as expected, shows less STL decrease than the case with SR, but wider frequency range where the STL increases than a poroelastic composite structure without resonators. This results from the fact that the frequency band of vibration suppression in the CR case is wider than in the SR case. As a result, to achieve the desired STL performance in a frequency range, the proposed composite structure using SR with tuned characteristic frequencies is enough; however, if a wider frequency band is expected even if there is a slight STL tradeoff, the CR case is a better option. Though the method proposed is only valid in the low-to-medium frequency range, the results obtained can benefit theoretical development of low-to-medium sound modulation applications, they are also valuable and illuminating for investigating the broadband sound modulation.

    • 多孔介质是一类利用内阻尼耗散能量进而达到减振降噪的材料, 由于造价低适用性好, 这类材料和弹性体组成的复合结构在建筑、机械和航空航天等领域都有广泛应用.

      在含多孔介质复合结构的建模中, 通常利用Biot理论[1]和等效流体模型[1]两类模型描述弹性波在多孔介质中的传播; 但是, 当考虑多孔介质固相骨架弹性特征时, 等效流体模型不再适用[1,2], 必须采用基于Biot理论的多孔介质模型. 这类采用Biot理论讨论的多孔介质夹层复合结构声振问题, 已经有很多研究[3-6]. 利用Allard等[7]和Deresiewicz[8]的简化模型, Bolton等[3]采用Biot理论对二维夹层板结构的随机声传播损失(STL)进行了研究, 分析了几种边界条件下的结果; 此后, Zhou等[4,9]、Liu等[5,10]以及Talebitooti等[11,12]也利用Biot理论对此类含多孔介质板壳夹层结构问题的三维情况进行了分析, 并讨论了外流等因素对STL的影响. 此外, 也有利用Biot模型结合数值方法进行这类复合结构问题研究[13-15]. 尽管这类含多孔介质复合结构具有优异声学性能, 对于中低频问题, 由于多孔介质在此频域效果有限, 并不能很好地提高复合结构的声学性能[6].

      由于具有优秀的中低频特性, 周期结构从20世纪50年代起就被广泛研究[16,17]; 近年来, 周期结构相关理论在声子晶体[18]及超材料研究[19]等领域得到了广泛应用. 研究表明, 周期结构对弹性波的调控机理有两种[20,21], 即布拉格散射和局域共振, 前者对于低频长波长弹性波适用性有限; 因此, 在中低频短波长弹性波调制中, 利用周期结构理论, 由弹性体结合局域共振系统(弹簧-振子系统、Helmholtz振子等)构成的具有反常特性的周期复合结构(“超材料结构”)得到了广泛研究和应用. 近年来, 此类周期复合结构在弹性波吸收[22]、声波调控[21,23-25]与振动抑制[26,27]方面有很多研究进展, 其中, 理论[23,24,27]、实验[23,27]和数值[22,26]方法都有应用. 尽管如此, 低频宽带轻质高效的声振抑制问题, 仍是理论与工程中有挑战性的问题[28].

      多孔介质是声振控制中的轻质高效方案, 而周期结构在中低频应用中有优势. 研究表明, 结合两种方案可以取得更好的声调控效果[29,30]; 然而, 相关研究目前还不充分[6,31-33], 其理论研究手段仍需要进一步发展; 二者的协同机理也需要进一步探究.

      为取得理想的中低频隔声特性, 本文结合含多孔介质复合板和周期振子系统, 组成了一种新型周期复合结构; 随后, 采用等效动态质量描述振子系统, 利用薄板理论和Biot多孔弹性方程建立了复合结构理论模型, 分析了其在中低频域的声振特性, 讨论了多孔介质、周期振子等对此复合结构STL的影响. 此外, 对隔振结构中常见的组合振子系统, 也讨论了其隔声可行性, 并与简单振子系统结果进行了对比分析. 值得注意的是, 结合周期振子和薄板结构的振动及声学特征研究国内外已有很多[34-37], 这里主要关注加入多孔介质后的复合结构隔声特性, 不再详细说明附周期振子薄板结构结果, 必要时直接引用对照.

    2.   复合结构及其相关理论和求解
    • 图1(a) 为含多孔介质周期复合结构示意图, 此结构由多孔介质、薄板和周期分布的弹簧振子系统构成; 弹簧振子系统沿$x$方向周期分布, 周期间距为$a$; 在每个周期内, 假设含有${N_s}$个振子(或振子系统), 第$i$个振子的刚度和质量分别为${k_i}$${m_i}$($i = 1,2, \cdots,{N_s}$). 假定复合结构在$y$方向无限延伸, 单位平面波以仰角$\varphi $入射, 其速度势为${\varPhi _i} = \exp ( {{\rm j}\omega t -} {{\rm j}{{kr}}} )$, 其中${{k}} = \left( {{k_x},{k_z}} \right)$为入射波矢, ${{r}} = (x,z)$为位置矢量, ${\rm{j}} = \sqrt { - 1} $. 下文中, 简谐时间项$\exp \left( {{\rm{j}}\omega t} \right)$都不再显式写出.

      Figure 1.  Schematic of the poroelastic composite structure and its substructures: (a) The poroelastic composite structure; (b) the equivalent model; (c) the OU boundary connection; (d) the OB boundary connection; (e) the forces in OU boundary case; (f) the forces in OB boundary case; (g) the forces in a simple spring-mass resonator.

      图1(c)(g) 给出了复合结构的连接边界条件和相应部分受力情况. 多孔介质和板结构连接类型采用文献[3]中的分类, 记O类型为多孔介质开放边界, 即多孔介质界面直接与外部声学域接触; U类型为多孔介质间隔边界, 即多孔介质界面与弹性体间存在间隙(例如, 薄空气层); B类型为多孔介质固定边界, 即多孔介质界面与弹性体固定粘连. OU和OB类型为几种类型边界条件的组合, 分别如图1(c)图1(d) 所示. 当多孔介质与弹性体间为固定边界时, 由于多孔介质固相骨架中正应力和剪应力同时存在, 此时弹性体还会受到面内外力和面外力矩作用, 图1(e)图1(f) 给出了OU, OB两种边界情况下薄板法向的受力示意图.

      除采用图1(a)所示简单振子系统外, 复合结构中的周期振子单元也可由图2(a)图2(b) 所示的组合振子系统构成, 其相应的刚度、阻尼和质量如图所示 (其中, 参数$k_n^i$, $\zeta _n^i$$m_n^i$中的$n = 1,2$为振子系统部件编号, $i$为周期单元序号). 采用周期振子系统排布时, ${N_s} \geqslant 1$, 可行的排布情况如图3(暂不考虑三类振子系统混合分布的情况). 为便于说明和分析, 下文采用表1中缩写形式描述图3所示振子系统分布情况.

      Figure 2.  Schematic of the composite-resonator-structure: (a) Composite resonator type A, two resonators placed in serial connection; (b) composite resonator type B, two resonators placed in composite connection.

      振子系统分布含义
      N1SR周期间隔内分布多个简单振子系统, 各振子系统${m_i}$和${f_i}$均相等
      NNSR周期间隔内分布多种简单振子系统, 各振子系统${m_i}$或${f_i}$不同
      N1CR周期间隔内分布多个组合振子系统, 各振子系统$m_n^i$和$f_n^i$均相等
      NNCR周期间隔内分布多种组合振子系统, 各振子系统$m_n^i$或$f_n^i$不同

      Table 1.  Abbreviations of the distribution of resonator systems and their meanings.

      Figure 3.  Schematic of the arrangement of periodic resonators: (a) An array of simple resonators, denoted as multiple resonators (N1SR, with constant ${m_i}$ and ${f_i}$) or multiple kinds of resonators (NNSR, with different ${m_i}$ and ${f_i}$); (b) an array of composite resonators, denoted as multiple resonators (N1CR, with constant $m_n^i$ and $f_n^i$) or multiple kinds of resonators (NNCR, with different $m_n^i$ or $f_n^i$). The area in the dash-line denotes the periodic lattice, in panel (b), the composite resonator can be type B in Fig. 2

    • 常见多孔介质固相骨架可以有弹性变形, 此时必须采用Biot理论[38]描述多孔介质中弹性波的传播. 考虑图1(a) 所示周期复合结构, 以多孔介质固相位移${{{u}}^{\rm{s}}} = \left( {u_x^{\rm{s}},u_z^{\rm{s}}} \right)$和液相位移${{{u}}^{\rm{f}}} = \left( {u_x^{\rm{f}},u_z^{\rm{f}}} \right)$ 描述的多孔介质中弹性波传播方程为[38]

      其中, $\rho _{11}^*$, $\rho _{12}^*$, $\rho _{22}^*$, $A$, $N$, $Q$$R$是多孔介质特性参数[3,6].

      平面波入射情况下, 二维多孔介质域中的场变量, 如应力和位移等, 都可以由6个待定未知谐波贡献量${C_i}$($i = 1,2, \cdots,6$)唯一确定[3,6], 详细的推导方法和表达式可参考文献[3, 6], 这里不再赘述.

    • 对于中低频问题, 关注的频域低于板临界频率(coincidence frequency), 故采用薄板理论描述弹性波在板中的传播[39]. 在外力和外力矩同时作用时, 薄板的面内振动和面外振动方程分别为[6]

      其中, ${\rho _p}$为薄板面密度(或等效密度); h为薄板厚度; ${D_p}$为薄板面内刚度; ${f_x}$为薄板面内力; D为薄板弯曲刚度; ${f_z}$${M_y}$分别为薄板所受外力和外力矩; uw分别为薄板面内位移和面外位移. (3)式中, 符号 ± 的选取和多孔介质与薄板的相对位置有关, 在边界条件式中指定[3,4]. 根据声波反射和透射的Snell定律[40], 薄板位移uw可表示为

      其中, ${U_1}$${W_1}$为可由复合结构边界条件确定的待定谐波幅值.

      根据文献[35, 41], 对中低频问题, 弹簧振子系统可以等效为板上的附加质量块, 其效果相当于增加了板密度, 相应结果与平面波展开法的结果完全一致, 但是计算量大为降低[35]. 因此, 这里采用等效介质模型[35]描述薄板-振子耦合结构, 其做法是将周期单元内的弹簧振子系统等效为附加在均匀板上的额外质量, 相当于改变了板密度. 对图1(a) 所示的简单振子系统, 假定第i个振子的恢复力为${f_i}$, $i = 1,2, \cdots,{N_s}$, 则

      其中, ${w_i}$为第i个振子的质量块位移. 记第i个弹簧振子动态质量为${m_{{\rm{eq}},i}}\left( \omega \right)$, 根据${f_i} = {m_{{\rm{eq,}}i}}\ddot w$, 则${m_{{\rm{eq}},i}}\left( \omega \right)$

      其中, ${m_i}$为振子质量, ${\omega _i}$为振子特征圆频率, ${\eta _i}$为弹簧阻尼系数; 此结果与文献[35, 41]一致. 根据(6)式, 对于单位宽度薄板-简单振子耦合结构, 其等效动态密度为

      其主结构的位移传递率(displacement transmissibility) ${T_i}\left( \omega \right) = {{{w_i}} / w}$

      对于图2(a)图2(b)两种组合振子系统情况, 相应的振子系统方程为

      其中, ${{x}} = {\left[ {{x_1},{x_2},{x_3}} \right]^{\rm{T}}} = {\left[ {{{{w_1}} / w},{{{w_2}} / w},{f / w}} \right]^{\rm{T}}}$; f为主结构(薄板)对振子系统作用力; F为右端参数项; 两类组合振子系统的质量矩阵M、阻尼矩阵C和刚度矩阵K均在附录A给出. 由(9)式可得

      这里${{H}} = {\left( {{{K}} + {\rm{j}}\omega {{C}} - {\omega ^2}{{M}}} \right)^{ - 1}}$. 根据位移传递率的定义, 主结构(薄板)和振子系统质量块$m_1^i$$m_2^i$间的位移传递率分别为${T_1}\left( \omega \right) = {x_1}$, ${T_2}\left( \omega \right) = {x_2}$; 薄板-组合振子耦合结构的等效动态质量为${m_{{\rm{eq}},i}}\left( \omega \right) = {{{{ - f} / {\ddot w}} = {x_3}} / {{\omega ^2}}}$, 相应单位宽度薄板-组合振子耦合结构的等效动态密度为

      其中, 上标A, B分别对应图2中复合振子系统(a)和(b); 式中, 上标A, B一一对应取得.

    • 复合结构周围的声学域由声速度势Φ描述, 它满足声学域波动方程

      其中, c为相应声学域中波速. 对于OU边界情况, 假定其入射侧声学域速度势$ {\varPhi _1} $, 多孔介质和薄板间空气层速度势$ {\varPhi _2} $和透射侧速度势$ {\varPhi _3} $分别如下:

      其中, ${R_1}$,${I_2}$,${R_2}$${T_3}$为待定谐波幅值, 由下文边界条件确定. 由(12)—(15)式可得

      其中, ${k_ \times } = {\omega / {{c_ \times }}}$; $k_z^ \times $${k_ \times }$分别为相应域的z向波数分量$k_z^i$, $k_z^a$$k_z^t$和相应域波数${k_i}$, ${k_a}$${k_t}$; ${c_ \times }$为相应域波速${c_i}$, ${c_a}$${c_t}$.

    • 在文献[3, 4]中, 多孔介质和相邻声学域或弹性体连接的O, U和B三种情况边界条件方程已经被详细讨论给出, 这里不再说明. 以OU边界情况为例, 以下给出与本文相关的复合结构边界条件来说明求解过程. 相关方程为

      其中, $\epsilon$为多孔介质孔隙率; ${\rho _i}$,${\rho _a}$${\rho _t}$分别为入射侧、空气夹层和透射侧声学域介质密度; s为多孔介质液相压力; ${\sigma _z}$${\tau _{zx}}$为多孔介质固相正应力和切应力; 式中方程 (i)—(xi) 都施加在复合结构中不同域交界面或薄板中面上.

      根据文献[3, 4, 6], 利用(4), (13)—(15)式, (17)式可整理为

      其中, 未知矢量${{x}} = {\left[ {C_1},{C_2},{C_3},{C_4},{C_5},{C_6},{W_1},{R_1},\right.}$$\left.{ {I_2},{R_2},{T_3}} \right]^{\rm{T}}$, 系数矩阵A和力向量f附录B中给出. 根据(18)式, 可求得${{x}} = {{{A}}^{ - 1}}{{f}}$, 由此可得声学域声速度势和多孔介质场及薄板位移.

      声学域声功率定义为

      其中, 声压$p = \rho {{\partial \varPhi } / {\partial t}}$, 速度${ v} = - \nabla \varPhi $, ${v^*}$为法向速度v的共轭复数. 声功率传递系数定义为$\tau = {{{I_{\rm{t}}}} / {{I_{\rm{i}}}}}$, ${I_{\rm{i}}}$${I_{\rm{t}}}$分别为入射声功率和透射声功率; 对于散射声场, 其随机传声损失(STL)为[3,35]

      根据(18)式的结果即可求得复合结构的声功率传递系数和STL.

    3.   模型验证及分析
    • 选取薄板、空气域及简单振子参数如表2所列.

      薄板 空气域(20 ℃, 1 atm) 简单振子
      h/mmE/GPaν${\rho _{\rm{p}}}$/kg·m–3 ρ/kg·m–3c0/m·s–1ha/mm fr/Hzγ
      1700.332700 1.204343.212 3000.2

      Table 2.  Parameters of plate and resonators.

      表2中, fr图1 (a)简单振子特征频率; $ \gamma =$ $ m_{\rm sum}/ \left( \rho _{\rm p}ha \right)$为振子系统总质量与薄板单位宽度总质量的比值, 以下简记为振子系统质量比. 多孔材料参数如表3所列.

      参数参数
      ${\rho _{\rm{s}}}$/kg·m–330 $\epsilon$0.9
      ${\rho _{\rm{f}}}$/kg·m–31.204 ${\tau _\infty }$7.8
      ${E_{\rm{s}}}$/MPa0.8(1+0.265${\rm{j}}$) ${\sigma _0}$/MKS rayls25000
      ${\nu _s}$0.4 hp/mm50

      Table 3.  Parameters of the porous media used here

      表3中, ${\rho _{\rm{s}}}$${\rho _{\rm{f}}}$分别为多孔介质固相和气相密度, ${E_{\rm{s}}}$${\nu _{\rm{s}}}$为其固相杨氏模量和泊松比, $\epsilon$为孔隙率, ${\tau _\infty }$为几何形状因子, ${\sigma _0}$为流阻, hp为多孔介质厚度. 采用一维Simpson积分计算复合结构的STL, 积分步长取为${{\text{π}} / {180}}$, 计算频域为1—1000 Hz. 下文如无特殊说明, 所有计算均采用上述计算设置.

    • 为验证本文的理论模型, 分别将其退化为文献[35]中附加亚波长周期振子的均匀薄板结构模型($\epsilon = 1$, ${h_{\rm{p}}} = {h_{\rm{a}}} = 0$, 图4(a)为随机入射情况, 图4(b)为斜入射情况)及文献[3]中多孔介质复合结构模型(${m_{{\rm{sum}}}} = 0$, 图4(c)), 并选用相同计算参数与相应结果进行对照, 验证算例模型示意图及其结果在图4中给出.

      Figure 4.  Validation of the results here with previous results: (a) The diffuse case in Ref. [35]; (b) the oblique incident cases in Ref. [35]; (c) the composite poroelastic structure without resonator in Ref. [3]. The lines are results obtained here, while the marks are the results in the references.

      图4可以看出, 对于附加周期振子均匀薄板和含多孔介质复合板两种情况, 本文方法计算得出的结果与相应文献中数据都符合良好, 表明本文方法是可靠的.

    • 保持振子系统质量比$\gamma = 0.2$, 取复合结构中周期简单振子质量块${m_0} = 27\;{\rm{g}}$ (N1SR情况), 有无多孔介质时两种边界条件下的STL情况见图5.

      Figure 5.  Influence of porous material on the STL of the multiple-single-type-resonator composite structure with different characteristic frequencies: (a) OU case; (b) OB case. The solid lines correspond to cases with porous materials.

      图5可以看出, 在OU和OB两种边界条件下, 附加多孔介质对复合结构STL影响情况基本一致. 采用某一特征频率振子时, 多孔材料的引入可以在远离振子特征频率频域提升其STL, 这是由多孔介质的声学性能决定的. 周期振子特征频率附近, 有无多孔介质对STL提升并不明显, 此区域STL主要由周期振子决定, 但由于多孔介质的引入, 在振子系统特征频率后的STL降低趋势被削弱. 因此, 多孔介质的引入, 总体上增大了结构的隔声能力, 这与文献[1]中对多孔介质的特性描述一致.

      图6给出了两种边界条件下含相同简单振子系统复合结构(fr = 300 Hz)有无多孔介质情况, 及相应不含振子复合结构的STL. 由图6可知, 采用某一特征频率振子时, 多孔材料的引入可以在远离振子特征频率频域提升其STL, 最终趋近于相应不含振子复合结构的STL. 采用多孔介质和简单周期振子复合结构, 可以使得结构的隔声性能在保有多孔介质宽频优势情况下, 在特定频点或频段有一定提升; 或保有简单周期振子系统特定频点或频段优势情况下, 在宽频域有一定提升.

      Figure 6.  The STL of multiple-single-type-resonator composite structure (fr = 300 Hz) with/without porous, and composite structure without resonators: (a) OU case; (b) OB case. Composite structure here with porous material: Porous + Resonator. Without porous material: Resonator. Composite structure without resonators: Porous.

    • 保持振子系统质量比$\gamma = 0.2$, 取复合结构中周期简单振子质量块${m_0} = $27 g, 采用不同特征频率简单振子系统时, 复合结构在OU和OB两类边界条件下的STL如图7(a)图7(b). 由图7可知, 采用不同特征频率的周期振子系统, 都使得在此特征频率附近STL先显著提升然后降低, 最后恢复到不含振子时STL的水平. 这表明, 合适引入和布置周期振子, 可以在一定频域内达到提升STL的目的.

      Figure 7.  Influences of resonators with different characteristic frequencies on the STL: (a) OU case; (b) OB case.

      图8(a)图8(b)给出了两种边界条件下, 由特征频率fr = 300 Hz的简单振子构成的复合结构的STL及其位移传递率Ti、振子动态质量meq和板等效动态密度${\rho _{{\rm{eq}}}}$. 从图8可以看出, 在振子特征频率附近Ti, meq${\rho _{{\rm{eq}}}}$变化趋势一致, 都由极大(f = 290 Hz)下降为极小(f = 307 Hz); STL也呈由极大(f = 298 Hz)到极小(f = 325或335 Hz)的变化趋势, 但STL变化与Ti, meq${\rho _{{\rm{eq}}}}$变化特征并不同步, 存在一定的频率滞后. 这是由于局域共振板在特征频率附近存在一个范围约为${f_{\rm{r}}}$${f_{\rm{r}}}{( {1 + }}$${{ \gamma } )^{1/2}}$的带隙[34,35], 而${f_{\rm{r}}}$处板等效动态密度呈极大值(无阻尼时为无穷大), ${f_{\rm{r}}}{\left( {1 + \gamma } \right)^{1/2}}$处呈极小; 此频域为质量定律控制区域[42], 其隔声量(即STL)与结构面密度呈正相关, 故该频域内STL呈极大过渡到极小趋势.

      Figure 8.  (a) STL of OU and OB case in periodically-arranged single simple resonator case, and its displacement transmissibility Ti; (b) equivalent mass meq of a single resonator and the dynamic density ${\rho _{{\rm{eq}}}}$ of the equivalent plate.

    • 不同振子系统的动态特性不同, 使用合适的组合振子系统可以一定程度上拓宽振子系统的作用频域, 进而达到更宽频带振动调控[43]. 为便于讨论, 令组合振子系统A, B中振子质量刚度分别满足如下关系$m_2^i = r \cdot m_1^i$, $k_2^i = s \cdot k_1^i$, 其阻尼比为$\eta _n^i =$$ {{\zeta _n^i} / {2m_n^i\omega _n^i}}$, $\omega _n^i = \sqrt {{{k_n^i} / {m_n^i}}} $, $n$ = 1, 2. 对无阻尼情况 (即$\zeta _1^i = \zeta _2^i = 0$), 组合振子系统A特征圆频率为

      (21)式与文献[43]中无阻尼组合振子结果一致. 同样, 无阻尼组合振子系统B的特征圆频率为

      由(21)和(22)式可知, 组合振子A和B二特征频率间的带宽为$\Delta {f^i} = {{\left| {\omega _1^i - \omega _2^i} \right|} / {2{\text{π}}}}$, i = A, B. 根据前述分析, 复合结构STL在振子特征频率附近呈由极大到极小变化趋势, 为获得较高STL, 组合振子二特征频率间带宽$ \Delta f ^i$应控制到合适水平, 相关研究还在进行中, 这里仅给出一些初步结果. 忽略弹簧质量, 保持振子系统质量比$\gamma = 0.2$, 取组合系统参数如表4所列.

      参数$\omega _1^i$$\omega _2^i$rs$\eta _1^i$$\eta _2^i$
      600${\text{π}}$10880.0750.06250.010.05

      Table 4.  Parameters of the composite resonators.

      组合振子系统 A和B 的位移传递率${T_1}$, ${T_2}$和动态质量${m_{{\rm{eq}}}}$变化情况见图9.

      Figure 9.  Displacement transmissibility and dynamic mass of the mass components in the two composite resonators: (a1) Displacement transmissibility ${T_1}$, ${T_2}$ of composite resonator type A; (a2) dynamic mass ${m_{{\rm{eq}}}}$ of composite resonator type A; (b1) displacement transmissibility ${T_1}$, ${T_2}$ of composite resonator type B; (b2) dynamic mass ${m_{{\rm{eq}}}}$ of composite resonator type B.

      图9(a1)图9(b1) 可知, 在组合振子系统A和B的共振频率附近, 系统中振子位移传递率均呈先增大至共振频率再降低的趋势, 且振子系统A和B两个共振频率之间形成了一段振动抑制频带, 此频带宽度与组合振子系统特征频率有关; 组合振子系统A和B均可以对主结构振动产生明显抑制, 但其抑制带宽不同. 由于声波通过此复合结构中薄板(主结构)向透射侧传播, 针对薄板向外部辐射声的振动模态采用合适的组合振子系统, 预计可以改善结构隔声性能.

      图10显示了保持振子系统质量比$\gamma = 0.2$, 单周期内分布${N_{\rm{s}}} = 4$个相同简单振子, 组合振子A或B和不含振子时复合结构的STL情况. 由图10可知, 在两种边界条件下, 由于振子系统特征频率附近其STL呈先升高后降低趋势, 对单一共振频率的简单振子系统, 其STL提升频域较窄; 选用合适参数的组合振子系统A或B, 相对简单振子系统, 可以在较宽频域提升复合结构的STL, 同时不显著降低其STL值.

      Figure 10.  STL of the proposed composite structure with 4 identical simple resonators (Single resonator), composite resonators of type A or B versus its STL without any resonators (Without resonator) in a periodic lattice: (a) OU boundary case; (b) OB boundary case.

    • 在周期单元内布置多种振子(或组合振子), 可以拓宽复合结构的振动抑制频带[35], 进而提高其STL水平. 以下保持振子系统质量比为0.2, 分析采用多种振子系统对复合结构的声调控效果.

    • 选取不同特征频率简单振子系统, 讨论以下两种情况复合结构的声调控特性: (A) 等质量振子, 振子质量$ {m_{i}} $为振子系统总质量${m_{{\rm{sum}}}}$${1 / {{N_{\rm{s}}}}}$; (B) 振子质量$ {m_{i}} $各不相同, $ {m_{i}} = {m_0} + (i - 1) $$\Delta m$, $i = 1,2, \cdots,{N_{\rm{s}}}$. 假定弹簧阻尼系数为$ {\eta _i} = $0.05. 图11给出了OU, OB两种边界情况下NNSR分布时情况A, B的结果, 其中振子特征频率分别为$f_{\rm{r}}^i = 300 + 50\left( {i - 1} \right)$ (Hz), $i = 1,2, \cdots,{N_{\rm{s}}}$, ${N_{\rm{s}}} = $4, 5, 6, 7.

      图11可知, 特征频率各不相同的多振子系统, 无论其各振子质量是否相等, 都能在由振子特征频率决定的频带提升复合结构STL, 此结论与文献[35]中相似; 同时, 非均匀质量分布的周期振子对复合结构的STL提升更为明显, 缺点是在振子特征频率决定的频带之外, 其STL下降也更为剧烈. 这是由振子质量的不均匀分布决定的. 此外, 两类边界情况下不含振子复合结构的STL变化情况不同, 此结果决定了复合结构STL的整体趋势. 若仅考虑某频率范围的隔声性能, 采用不同振子, 按照振子特征频率依次增大, 振子质量依次递增布置方案, 可以提升对应频率范围的隔声性能.

      Figure 11.  STL of the composite structure with NNSR configuration under two boundary cases: (a1), (a2) Case A; (b1), (b2) case B, ${{\Delta m} / {{m_{{\rm{sum}}}}}} = 0.04$. Here (a1) and (b1) correspond to OU case, (a2) and (b2) correspond to OB case.

    • 忽略组合振子中的弹簧质量, 保持振子系统质量比$\gamma = 0.2$, 选取组合振子系统参数如表5所列.

      参数rs$\eta _1^i$$\eta _2^i$
      0.0450.0400.010.05

      Table 5.  Parameters of the composite resonators.

      表5中, 各组合振子特征圆频率$\omega _1^i = 2{\text{π}}f_1^i$, $f_1^i = f_1^0 + 50(i - 1)$(Hz), $i = 1,2, \cdots,{N_{\rm{s}}}$, ${N_{\rm{s}}} = 4$. 在周期间隔内, 选取不同特征频率的组合振子A或B 时, 复合结构STL情况见图12.

      Figure 12.  STL of the proposed composite structure under NNCR configuration: (a1), (a2) Composite resonator type A; (b1), (b2) composite resonator type B. Here (a1) and (b1) correspond to OU case, (a2) and (b2) correspond to OB case.

      图12可知, 在单个周期内引入多个不同特征频率的组合振子, 同简单振子情况类似, 可以提高复合结构在其特征频率决定的频域内的STL, 在此频域内, OU和OB两种边界条件下呈现了类似的STL提升趋势; 在越过该频域后, 其STL相对不含振子复合结构有降低, 降低的幅度和复合结构边界条件及振子类型有关. 此特性与简单振子情况类似.

    • 为对照不同振子系统分布对复合结构STL的影响, 保持振子系统质量比$\gamma = 0.2$, 取${N_{\rm{s}}} = 4$, 各振子系统特征频率$ {f_r^i} $(简单振子)或$f_1^i$(组合振子)取为300+50(i–1) (Hz), i = 1, 2, ···, Ns. 不同周期振子系统分布时复合结构的STL情况如图13所示.

      Figure 13.  STL of different resonator system configuration: (a), (a1) OU case; (b), (b1) OB case. $\Delta m = 0$ and $\Delta m > 0$ correspond to simple resonator case NNSR. Type A and Type B correspond to composite resonator case NNCR.

      图13可知, 几类周期振子系统对复合结构STL的提升都表现为振子系统特征频率附近频域的局部提升, 在越过该频域后, 振子系统会降低复合结构的STL, 远离复合结构特征频率后恢复到无振子情况STL值. 对比组合振子系统A, B和简单振子系统 ($\Delta m = 0$$\Delta m > 0$) 结果, 简单振子系统总体上可以取得不错的STL, 组合振子系统并不能进一步显著提升复合结构STL, 只在STL提升带宽上有优势. 在合适的振子参数下, 相对简单振子系统, 组合振子系统可拓展复合结构的STL提升带宽, 且不显著降低复合结构STL水平.

      综合简单振子系统和组合振子系统结果, 在实际工程问题中, 采用简单振子系统预期可以取得合适的STL提升, 如果要进一步增大STL提升带宽, 可以考虑使用组合振子.

    4.   结 论
    • 含多孔介质复合结构和局域共振周期结构分别有各自的声学优势, 然而目前对两者协同效应的研究尚不充分. 为取得理想的中低频隔声性能, 本文提出了一种结合以上两类声振抑制方案的复合结构模型, 并在中低频域对其STL进行了理论求解和分析. 研究表明, 合适地引入和布置周期振子, 可以在一定频域内提升复合结构的STL. 这是因为附加的振子系统可以对主结构(薄板)振动产生明显抑制, 合适的振子系统参数, 可以抑制某些振动模态的声辐射, 进而提升其STL水平. 对比简单振子和组合振子的结果可知, 在实际问题中, 若要提升结构在某段频域的STL, 可选取相应特征频率的简单振子系统, 按照非均匀质量周期布置; 如需进一步增大其STL提升带宽, 可考虑使用更为复杂的组合振子系统. 尽管本文方法只适用于中低频问题[35], 这些研究结果可以为宽频减振降噪研究和中低频减振降噪提供理论参考.

    附录A.   (9)式中振子系统相关矩阵
    • A类组合振子系统的质量矩阵M、阻尼矩阵C和刚度矩阵K

      其右端项${{F}} = {\left[ {0, - k_1^i - {\rm{j}}\omega \zeta _1^i,k_1^i + {\rm{j}}\omega \zeta _1^i} \right]^{\rm{T}}}$.

      B类组合振子系统的质量矩阵M、阻尼矩阵C和刚度矩阵K

      其右端项F

    附录B.   (18)式中系数矩阵及力矢量
    • ${L_1} = {h_{\rm{p}}} + {h_{\rm{a}}}$, ${L_2} = {h_{\rm{p}}} + {h_{\rm{a}}} + \displaystyle\frac{{{h_1}}}{2}$${L_3} = {h_{\rm{p}}} +$$ {h_{\rm{a}}} + {h_1}$, 式中系数矩阵A中的非零项为

      这里, 参数${Q_0}$, ${R_0}$, ${b_1}$, ${b_2}$, ${N_0}$, $g$, ${k_{1z}}$, ${k_{2z}}$, ${k_{3z}}$, ${k_1}$, ${k_2}$${k_3}$都在文献[3]中给出.

      力向量f中非零项为$f\left( 1 \right) = {\rm{j}}\omega {\rho _i}\epsilon$, $f\left( 2 \right) = {\rm{j}}\omega {\rho _i}\;\times$$\left( {1 -\epsilon} \right)$, $f(3) = {\rm{j}}k_z^i$.

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